# Hopf algebras in algebraic topology

I’m going to talk today about a little bit of algebra. There’s a beautiful relation to algebraic topology which I’ll touch on as well. (For a brief introduction to cup products, refer to section 7 of my notes here: http://www.mit.edu/~sanathd/papers/algebraic-topology.pdf.)

Definition: A Hopf algebra is a graded $R$-algebra $A = \bigoplus_{n=0}A^n$ satisfying:

1. There is an identity $1\in A^0$ such that $R\to A^0$ given by $r\mapsto r\cdot 1$ is an isomorphism.
2. There is a homomorphism of graded algebras $\Delta:A\to A\otimes_R A$, called the diagonal/coproduct, such that for $\alpha\in A^n$ and $n>0$, we have: $\displaystyle\Delta(\alpha) = \alpha\otimes 1 + 1\otimes \alpha + \sum_{i=1}^{n-1}\beta_i\otimes\gamma_{n-i}$, where $\beta_k,\gamma_k\in A^j$.

For example: $R[\alpha]$. Because elements of $R[\alpha]$ of dimension lower than $\alpha$ are the elements of $R$, the terms $\beta_k$ and $\gamma_k$ are $0$, and the diagonal is hence $\Delta(\alpha) = \alpha\otimes 1 + 1\otimes \alpha$.

Another example is the algebra $k[\alpha]/(\alpha^n)$ where $k$ is a field. In this case, $\alpha$ is primitive, i.e., $\Delta(\alpha) = \alpha\otimes 1 + 1\otimes\alpha$. Therefore, we have:

$\displaystyle\Delta(\alpha^n) = \alpha^n\otimes 1 + 1\otimes \alpha^n + \sum_{i=1}^{n-1}\binom{n}{i}\alpha^i\otimes\alpha^{n-i}$

But because we’re thinking of the truncated polynomial ring, $\alpha^n = 0$, and hence:

$\displaystyle\sum_{i=1}^{n-1}\binom{n}{i}\alpha^i\otimes\alpha^{n-i} = 0$

Thus if $0, we must have $\binom{n}{i} = 0$. If $k$ is of characteristic zero this is impossible. If $k$ has nonzero characteristic $p$, then $n$ must be a power of $p$ (this can be proved using the fact that the freshman’s dream is true in characteristic $p$).

Yet another example is the exterior algebra $\Lambda_R[\alpha]$ over $R$ with an odd-dimensional generator. (This is the free $R$-module with basis generated by $\alpha$ with multiplication defined by $\alpha^2 = 0$. For example, the cohomology ring $\mathrm{H}^\ast(T^1;R)\cong\Lambda_R[\alpha]$.) This is a Hopf algebra with diagonal $\Delta(\alpha) = \alpha\otimes 1 + 1\otimes\alpha$; to show that it indeed forms a Hopf algebra we must only check that $\Delta(\alpha^2)=0$. But $\Delta(\alpha^2) = \Delta(\alpha)^2$, so consider $(\alpha\otimes 1 + 1\otimes\alpha)^2$. This is equal to $\alpha^2\otimes 1 + 1\otimes\alpha^2 + (\alpha\otimes 1)(1\otimes\alpha)$. The latter term is $0$ since $\alpha$ has odd dimension, and $\alpha^2 = 0$ by definition, so we’re done.

One more rather important example: if $A$ is a finitely generated free $R$-module in each dimension then $A^\ast$ is a Hopf algebra, called the dual Hopf algebra (see Proposition 3C.10 in Hatcher). The product $A^\ast\otimes A^\ast\to A^\ast$ is the dual of the diagonal $\Delta:A\to A\otimes A$, and the diagonal $A^\ast\to A^\ast\otimes A^\ast$ is the dual of the product $A\otimes A\to A$.

Ok, so now onto topology. It turns out that there are natural examples of Hopf algebras in algebraic topology. We need two pieces of background. First, the notion of a H-space. Recall the notion of a topological group; this is a space $X$ with a group structure such that the product $X\times X\to X$ and the inverse map $X\to X$ are continuous. Every Lie group is a topological group, for example. This motivates:

Definition: A H-space is a space with a continuous multiplication $\mu:X\times X\to X$ and identity $e\in X$ such that $x\mapsto\mu(x,e)$ and $x\mapsto \mu(e,x)$ are homotopic to the identity.

For example, every topological group is a H-space. Another classic and very important example is a loop space $\Omega X$, whose product is:

$\displaystyle(\mu(\alpha,\beta))(t) = \begin{cases} \alpha(2t) & 0\leq t\leq 1/2\\ \beta(2t-1) & 1/2\leq t\leq 1 \end{cases}$
where $\alpha,\beta\in\Omega X$

Note that this isn’t necessarily commutative. Now, suppose $X$ is a H-space such that:

1. $\mathrm{H}^0(X;R)\cong R$, which is implied, for example, if $X$ is path-connected.
2. $\mathrm{H}^i(X;R)$ is finitely generated for each $i$, so the cross product $\mathrm{H}^\ast(X;R)\otimes_R\mathrm{H}^\ast(X;R)\to \mathrm{H}^\ast(X\times X;R)$ is an isomorphism.

The multiplication $\mu:X\times X\to X$ induces $\mu^\ast:\mathrm{H}^\ast(X;R)\to \mathrm{H}^\ast(X\times X;R)$, giving a map:

$\displaystyle \mathrm{H}^\ast(X;R)\xrightarrow{\Delta}\mathrm{H}^\ast(X;R)\otimes_R\mathrm{H}^\ast(X;R)$

This is a map of $R$-algebras. Let $i:X\to X\times X$ be the inclusion $x\mapsto (e,x)$, and consider the commutative diagram:

The map $F$ satisfies $F(\alpha\otimes 1)=\alpha$ and $F(\alpha\otimes \beta) = 0$ for $|\beta|>0$. By commutativity, $\mu i \simeq 1$, and hence $F\Delta = 1$. Hence the component of $\Delta(\alpha)$ in the tensor $H^0(X;R)\otimes H^n(X;R)$ is $1\otimes \alpha$. Continuing like so, we get the following formula for $\Delta(\alpha)$ where $\beta_i,\gamma_i$ are of degree $i$:

$\displaystyle \Delta(\alpha) = 1\otimes \alpha + \alpha\otimes 1 + \sum_{i=1}^{n-1}\beta_i\otimes \gamma_{n-i}$

And we get the structure of a Hopf algebra! This is very cool. Let’s use the examples of Hopf algebras we computed early on to look at examples of H-spaces. So for finite $n$, we see that $\mathbf{CP}^n$ is not a H-space by the example above about truncated polynomial rings. But, because polynomial rings are Hopf algebras, $\mathbf{CP}^\infty$ is a H-space! Recall from homology:

Proposition: The boundary map in $C_\ast(X\times Y)$ is given by $d(e^i\times e^j) = de^i\times e_j + (-1)^ie^i\times de_j$. The product of a boundary and cycle is a boundary because $da\times b = d(a\times b)$ and $a\times db = (-1)^id(a\times b)$ if $da=0$. This gives a bilinear map $\mathrm{H}_i(X;R)\times \mathrm{H}_j(Y;R)\to \mathrm{H}_{i+j}(X\times Y;R)$, and hence a homomorphism $\mathrm{H}_i(X;R)\otimes \mathrm{H}_j(Y;R)\to \mathrm{H}_{i+j}(X\times Y;R)$, which is the cross product in homology.

If $X$ is a H-space, the composition $\mathrm{H}_\ast(X;R)\otimes \mathrm{H}_\ast(X;R)\to \mathrm{H}_\ast(X\times X;R)\to \mathrm{H}_\ast(X;R)$ is called the Pontryagin product. This isn’t associative unless the product $\mu:X\times X\to X$ is associative up to homotopy. We then have a natural example of dual Hopf algebras coming from algebraic topology (the result follows from the fact that the hypotheses imply that $\mathrm{H}^n(X;R)\cong \mathrm{Hom}_R(\mathrm{H}_n(X;R),R)$.)!

Theorem: If $X$ is a H-space whose multiplication is associative up to homotopy and $\mathrm{H}^i(X;R)$ is a finitely generated free $R$-module for all $i$, then $\mathrm{H}_\ast(X;R)$ with the Pontryagin product is the dual Hopf algebra of $\mathrm{H}^\ast(X;R)$.

What about $\mathbf{RP}^n$; is it a H-space? If we choose $k=\mathbf{Z}/2\mathbf{Z}$ in the example about truncated polynomial rings, we see that we must have $n+1$ a power of $2$. The universal cover of a path-connected H-space is a H-space, and Adams showed that $S^1,S^3,S^7$ are the only spheres which are H-spaces. Since $\mathbf{RP}^n=S^n/{\pm 1}$ for $n=1,3,7$ gets their H-space structures from $S^1,S^3,S^7$, and thus $\mathbf{RP}^n$ can be a H-space only for $n=1,3,7$. A pretty cool result, huh? This depends on Adams’ theorem (the Hopf invariant one theorem) on the values of $n$ for which $S^n$ has a H-space structure, which was proved using K-theory! I plan on writing a post about this in the near future.

Best,
SKD